一階述語論理における代数的証明原理

抄録

A module D has been established, which has such laws of composition that correspond to sound deductions for predicate logic of first order. One major result of this theory is a new approach to the automated theorem proving, named algebraic proving principle. According to this principle, a set of Horn clauses is unsatisfiable if and only if a linear equation (proof equation) has a nonnegative solution. This implies that the laws of composition are complete as deduction if only Horn clauses are in concern. The elements of D are named sentences. The proof equation has a form which equates a specific sentence with a linear combination of sentences corresponding to the clauses, having the elements of some basic ring R of D as the unknown coefficients. In the case of propositional logic, the proof equation can be reduced to a simultaneous linear equation on Z (integer), and can be solved numerically. But in the case of predicate logic, D is a torsion R-module, and the problem of efficiently solving the proof equation is still open.